Discontinuous Galerkin Methods for Extended Hydrodynamics.

Abstract

This dissertation presents a step towards high-order methods for continuum-transition flows. In order to achieve maximum accuracy and efficiency for numerical methods on a distorted mesh, it is desirable that both governing equations and corresponding numerical methods are in some sense compact. We argue our preference for a physical model described solely by first-order partial differential equations called hyperbolic-relaxation equations, and, among various numerical methods, for the discontinuous Galerkin method. Hyperbolic-relaxation equations can be generated as moments of the Boltzmann equation and can describe continuum-transition flows.

Two challenging properties of hyperbolic-relaxation equations are the presence of a stiff source term, which drives the system towards equilibrium, and the accompanying change of eigenstructure. The first issue can be solved by an implicit treatment of the source term. To cope with the second difficulty, we develop a space-time discontinuous Galerkin method, based on Huynh’s “upwind moment scheme.” It is called the DG(1)–Hancock method.

The DG(1)–Hancock method for one- and two-dimensional meshes is described, and Fourier analyses for both linear advection and linear hyperbolic-relaxation equations are conducted. The analyses show that the DG(1)–Hancock method is not only accurate but efficient in terms of turnaround time in comparison to other semiand fully discrete finite-volume and discontinuous Galerkin methods. Numerical tests confirm the analyses, and also show the properties are preserved for nonlinear equations; the efficiency is superior by an order of magnitude.

Subsequently, discontinuous Galerkin and finite-volume spatial discretizations are applied to more practical equations, in particular, to the set of 10-moment equations, which are gas dynamics equations that include a full pressure/temperature tensor among the flow variables. Results for flow around a micro-airfoil are compared to experimental data and to solutions obtained with a Navier–Stokes code, and with particle-based methods. While numerical solutions in the continuum regime for both the 10-moment and Navier–Stokes equations are similar, clear differences are found in the continuum-transition regime, especially near the stagnation point, where the Navier–Stokes code, even when implemented with wall-slip, overestimates the density.